Frobenius splitting of Schubert varieties of semi-infinite flag manifolds

Abstract

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field ${\mathbb K}$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind-)scheme structure, its projective coordinate ring has a $\mathbb {Z}$ -model and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. 371 no.2 (2018)]) when $\mathsf {char}\, {\mathbb K} =0$ or $\gg 0$ , and the higher-cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$ . Some particular cases of these results play crucial roles in our proof [47] of a conjecture by Lam, Li, Mihalcea and Shimozono [60] that describes an isomorphism between affine and quantum K-groups of a flag manifold.

A presentation of the torus-equivariant quantum $K$-theory ring of flag manifolds of type $A$, Part I: the defining ideal

“…3.1 The H-equivariant K-group of semi-infinite flag manifolds. Let Q rat G denote the semi-infinite flag manifold, which is the reduced ind-scheme whose set of C-valued [Kat2] for details), where G is a connected, simply-connected simple algebraic group over C, B = HN is a Borel subgroup, H is a maximal torus, and N is the unipotent radical of B. One has the semi-infinite Schubert (sub)variety…”

“…In [Kat1,Kat3], Kato established an R(H)-module isomorphism Φ from QK H (G/B) onto the H-equivariant K-group K H (Q G ) of the semi-infinite flag manifold Q G , in which tensor product operation with an arbitrary line bundle class is induced from that in K H×C * (Q G ) by the specialization q = 1. In our notation, the R(H)-module isomorphism Φ sends the (opposite) Schubert class e…”

“…We then transfer it to an identity in QK H (F l n+1 ) through the R(H)-module isomorphism Φ from QK H (F l n+1 ) onto K H (Q G ) respecting the quantum multiplication ⋆ and the tensor product ⊗ with respect to the line bundle classes associated to anti-dominant fundamental weights −̟ i , 1 ≤ i ≤ n, which was obtained in [Kat1,Kat3] (see Section 5 for details).…”

“…Through the R(H)-module isomorphism Φ : QK H (F l n+1 ) → K H (Q G ), established in [Kat1,Kat3], which respects the quantum multiplication ⋆ with the line bundle class [O F l n+1 (−̟ i )] and the tensor product ⊗ with the line bundle class [O Q G (w • ̟ i )] for 1 ≤ i ≤ n, we can transfer the identity above in K H (Q G ) to the one below in QK H (F l n+1 ). For this purpose, we note that in our notation, the R(H)-isomorphism Φ sends the (opposite) Schubert class e µ [O w ][Q ξ ] to the semi-infinite Schubert class e −µ [O Q G (wt ξ ) ] for µ ∈ P , w ∈ W = S n+1 , and ξ ∈ Q ∨,+ , where…”

Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, II: arbitrary weights in ADE type

“…Let G be a simply-connected simple algebraic group over C with Borel subgroup B = HN , maximal torus H, unipotent radical N , Weyl group W , weight lattice P = i∈I Z̟ i , root lattice Q = i∈I Zα i , and coroot lattice Q ∨ = i∈I Zα ∨ i . The work [KoNOS] initiated the study of inverse Chevalley formulas in the equivariant K-group K H×C * (Q rat G ) of the semi-infinite flag manifold Q rat G associated with G, where the semi-infinite flag manifold Q rat G is a reduced indscheme whose set of C-valued points is G(C((z)))/(H(C)•N (C((z)))) (see [Kat2] for details), with the group C * acting on Q rat G by loop rotation; note that our K-group K H×C * (Q rat G ) is a variant of the Iwahori-equivariant K-group of Q rat G introduced in [KaNS]. The K-group K H×C * (Q rat G ) has a topological K H×C * (pt)-basis consisting of Schubert classes {[O Q G (x) ]} x∈W af indexed by the affine Weyl group W af = W ⋉ Q ∨ , where K H×C * (pt) ∼ = Z[q ±1 ][P ], with K H (pt) = R(H) ∼ = Z[P ] = Z[e λ | λ ∈ P ] the character ring of H and K C * (pt) = R(C * ) ∼ = Z[q ±1 ]; here q ∈ R(C * ) denotes the character of loop rotation.…”